On the topic of "test them all", I wrote this in my paper about error bounds and worst cases for complex multiplication (http://www.daemonology.net/papers/complexmultiply.pdf):

[For finding worst-case inputs] the approach taken was to perform an exhaustive search, taking several hours on the second author’s laptop, of IEEE single-precision inputs, using only a few arguments from Theorem 1 to prune the search.

Even though (2^32)^4 was far too large for it to be tractable to test all 4-tuples of single-precision inputs, a "pruned" search was able to easily cover and/or exclude the entire search space.